Element Selection for Partial Adaptive Nulling
Randy L. Haupt
Applied Research Laboratory, Pennsylvania State University, State College, PA
16801
E-mail: rlh45@psu.edu
Introduction
Partial adaptive nulling selects a subset of all the elements in an array to have
adaptive weights. Making only a few of the elements adaptive requires less
hardware and reduces computational complexity compared to fully adaptive
arrays. Partial adaptive nulling was first advocated in the 1980's [1] and is
important in power minimization adaptive algorithms [2]. This paper examines
the effects of adaptive element selection for a partially adaptive array.
Null Synthesis with a Subset of Elements
When N a elements of an N element array are adaptive, then the adaptive weights,
we ( i) = Δ e ( i ) e
jδ e( i )
, can be adjusted in phase and/or amplitude to place nulls in the
array factor. At the null locations, the array factor is zero [3].
N
∑
n =1
N
a
jk e( v ) −1) d cos φm
an e jk ( n −1)d cosφm − ∑ we( v ) e (
=0
(1)
v =1
Quiescent Pattern
Cancellation Pattern
Rewriting this equation into matrix form Ax = b yields
jk ( e(1) −1) d cos φ1
⎡e
⎢
⎢
⎢ jk ( e(1)−1) d cosφM
⎢⎣e
e
e
jk ( e( N a ) −1) d cos φ1
jk ( e ( N a ) −1) d cos φM
⎡ N
jk n −1 d cos φ
an e ( ) 1
⎤ ⎡ we(1) ⎤ ⎢ ∑
n =1
⎥⎢
⎥ ⎢
⎥⎢
⎥=⎢
⎥ ⎢w
⎥ ⎢N
⎥⎦ ⎣ e( Na ) ⎦ ⎢ ∑ a e jk ( n −1)d cosφM
⎢⎣ n =1 n
⎤
⎥
⎥
⎥
⎥
⎥
⎥⎦
(2)
where
e = vector containing indexes of the adaptive elements
φm = location of null m
an = quiescent amplitude weight of element n
d = element spacing
The least squares solution to (2) is
w = A† ( AA† ) b
−1
(3)
Fig. 1 is a plot of the cancellation patterns for a null at φ = 106.25o in the array
factor of a 32 element uniform array with λ / 2 element spacing with 8 adaptive
978-1-4244-4968-2/10/$25.00 ©2010 IEEE
elements. The shape of the cancellation beam depends upon which elements are
adaptive.
x
20
x
20
106.25
106.25
Quiescent
Quiescent
0
Gain (dB)
Gain (dB)
0
-20
-40
-40
-60
0
-20
-60
45
90
φ (degrees)
135
180
0
(a) e=1,2,3,4,29,30,31,32
180
106.25
Quiescent
Quiescent
0
Gain (dB)
0
Gain (dB)
135
x
20
106.25
-20
-40
-20
-40
-60
0
90
φ (degrees)
(b) e=13,14,15,16,17,18,19,20
x
20
45
-60
45
90
φ (degrees)
135
(c) e=1,5,9,13,17,21,25,29
180
0
45
90
φ (degrees)
135
180
(d) e=2,8,13,16,18,23,24,30
Fig. 1. Cancellation beams for 4 different selections of adaptive elements that
place a null at φ = 106.25o .
Adaptive element location determines the shape of the cancellation pattern. In
turn, the cancellation pattern shape determines the distortion to the adapted
pattern. If the adaptive elements are equally spaced, then grating lobes can form
as shown in Fig. 1c. When the elements appear in one clump as in Fig. 1b, then
the cancellation pattern main beam is wide compared to the cancellation pattern
main beam associated with adaptive edge elements (Fig. 1a). Random spacing
results in a cancellation beam with high sidelobes (Fig. 1d).
Element Selection for Partial Adaptive Nulling
As with null synthesis, the location of the adaptive elements in an array
determines the shape and gain of the cancellation pattern. Fig. 2 shows the
adapted, quiescent, and cancellation patterns for a 32 element dipole array
( d = λ / 2 ) when elements 1,2,3,4,29,30,31,and 32 are adaptive for both phase-
only adaptive nulling. The array was modeled using the method of moments [4].
The cancellation beam is found by subtracting the quiescent pattern from the
adapted pattern. No constraints were placed on the element weights, so the
distortion to the adapted pattern is significant.
x
20
Gain (dB)
106.25
122.5
0
-20
-40
0
45
90
φ (degrees)
135
180
Fig. 2. Adapted, quiescent, and cancellation patterns for the 32 element dipole
array when elements 1,2,3,4,29,30,31,and 32 are phase-only adaptive.
The unwanted pattern distortion in the previous example can be controlled by
placing limits on the adaptive weights or using fewer adaptive elements. Fig. 3
repeats the previous case but limits the maximum phase shift at the adaptive
elements to 90o . These limits result in much less distortion to the adapted pattern
while still placing the desired nulls.
x
20
Gain (dB)
106.25
122.5
0
-20
-40
0
45
90
φ (degrees)
135
180
Fig. 3. Results for the case in Fig. 2(a) when the adapted phase is limited to 90o .
Reducing the number of adaptive elements also reduces the resulting distortion to
the adapted pattern. For example, Fig. 4 is the adapted pattern with its
cancellation pattern superimposed on the quiescent pattern when 4 elements are
adaptive: 1, 2, 31, 32 with 8 bits of phase. The nulls are placed in the desired
directions by a very broad cancellation pattern. It intersects and nulls the
quiescent pattern at φ = 106.25o and φ = 136.25o . Phase-only nulling causes more
pattern distortion than using complex weights. Low amplitude weights at the edge
elements limit the possible pattern distortion by the cancellation beam.
x
20
106.25
122.5
Gain (dB)
0
-20
-40
-60
0
45
90
φ (degrees)
135
180
Fig. 4. Phase-only adaptive nulling using 4 edge elements.
Conclusions
The cancellation patterns were found analytically for array factors synthesized
from isotropic point sources and numerically for adaptive dipole arrays.
Decreasing the number of adaptive elements and/or decreasing the range of the
adaptive weights results in more pattern distortion. This approach can be used to
find the cancellation patterns of experimental adapted patterns.
References
[1] D. Morgan, “Partially adaptive array techniques,” Antennas and Propagation,
IEEE Transactions on, vol. 26, no. 6, 1978, pp. 823-833.
[2] R. L. Haupt, "Phase-only adaptive nulling with a genetic algorithm," IEEE
AP-S Trans., vol. 45, Jun 1997, pp. 1009-1015.
[3] H. Steyskal, R. Shore, and R. Haupt, “Methods for null control and their
effects on the radiation pattern,” Antennas and Propagation, IEEE
Transactions on, vol. 34, no. 3, 1986, pp. 404-409.
[4] FEKO Suite 5.4, EM Software and Systems (www.feko.info), 2008.